# The Real Term Premium in a Stationary Economy with Segmented Asset Markets

**Abstract**

This article proposes a general equilibrium model to explain the positive and sizable term premia implied by the data. The authors introduce a slow mean-reverting process of consumption growth and a segmented asset-market mechanism with heterogeneous trading technologies into an otherwise standard heterogeneous agent general equilibrium model. First, the slow mean-reverting consumption growth process implies that the expected consumption growth rate is only slightly countercyclical and the process can exhibit near-zero first-order autocorrelation, as observed in the data. This slight countercyclicality suggests that long-term bonds are risky, and hence the term premia should be positive. Second, the segmented asset-market mechanism amplifies the magnitude of the term premia because aggregate risk is highly concentrated in a small fraction of marginal traders who demand high compensation for taking risk. For sensitivity analysis, the role of each assumption is further investigated by removing each factor one at a time.

**INTRODUCTION**

The positive and sizable term premia observed in the data have been hard to reconcile using a standard structural macroeconomic model. Backus, Gregory, and Zin (1989) demonstrate the failure of a standard model in accounting for the sign and the magnitude of real bond risk premia. Campbell (1986), Donaldson, Johnsen, and Mehra (1990), and den Haan (1995) also experience the same difficulty with standard macroeconomic models.

Although equilibrium models are difficult to work with and have limited success, it is still important to try to understand the fundamental mechanisms behind positive and sizable term premia. For macroeconomists, the disconnect between the observed term premia in the data and what a standard structural macroeconomic model predicts is often referred to as the "term premium puzzle." The issue is also important to central bankers. As pointed out by Wright (2011), the term premia represent the relationship between the short rate, which is controlled by central banks, and the long rate, which relates more deeply to real economic activities. Hence, understanding the term premia helps central bankers evaluate the effectiveness of monetary policy and the mechanism behind its effects on the real economy. Finally, for investors, it is of utmost importance to understand term premia—to hedge against interest rate risk.

A standard macroeconomic model with the pricing kernel or the stochastic discount factor derived from a utility maximization problem generally has great difficulty matching the slope and the level of the term structure. Campbell (1986) shows that the term premium depends on the nature of the consumption growth process. If the consumption growth process is positively autocorrelated, then the expected future growth rate falls and bond prices rise in a recession. The long-term bond then becomes a good hedge, and hence the term premium is negative. On the other hand, if the consumption growth rate is negatively autocorrelated, then the model predicts a positive term premium since the long-term bond becomes risky because of its procyclical pricing. This intuition together with near-zero autocorrelation of consumption growth, that is, a random walk in empirical studies, implies that the term premium should be close to zero when the pricing kernel is derived from a standard macroeconomic model.

In addition, it is also well known that the pricing kernel of a standard model, which relies purely on the expected aggregate consumption growth rate, is not volatile enough to deliver a high market price of risk. Therefore, the standard model not only fails to match the sign of the term premium but also fails to generate the correct magnitude of the term premium.

In this article, we assume that the aggregate consumption process is trend stationary with a long memory process, which shows near zero but slightly negative autocorrelation of the consumption process. This consumption process alone generates positive term premia but with very small magnitude. This process is not easily statistically distinguished from a difference-stationary process such as the random walk. This view is supported by Christiano and Eichenbaum (1990,) who argue that no clear statistical evidence exits to support either a trend-stationary or a difference-stationary process of aggregate consumption. More specifically, we consider a slow trend-reverting aggregate consumption process in our model economy and hence the level of consumption can be well above or below its long-run trend for an extended period. With this process, when a bad shock is realized, the expected growth rate of consumption is only slightly higher because of its slow mean-reverting property. Therefore, the expected growth rate of consumption is only slightly countercyclical and the autocorrelation of consumption growth between two consecutive periods could be very close to zero but slightly negative (only –0.02 in our calibrated model), which is consistent with the random-walk-like consumption process in the data.

Following Chien, Cole, and Lustig (2011), there is a segmented asset-market mechanism in our model. Specifically, the model features a large fraction of households who do not participate in the equity market and hence do not bear any aggregate risk. There is, however, a small fraction of households who do participate in the equity market and hence bear a great amount of aggregate risk, which in this article results in a high market price of risk. In equilibrium, those households demand high risk compensation. As a result, high risk premia are obtained not only in equities but also reflected onto long-term bonds. Therefore, the segmented market mechanism amplifies the size and the magnitude of the term premia.

Our calibrated model considers not only the segmented asset-market mechanism but also the asymmetric bond positions of U.S. households. The data on U.S. households show that a large fraction of households carry long-term mortgage loans but save in short-term risk-free assets, such as checking or savings accounts. In other words, these households essentially borrow in long-term bonds by using housing as a collateral and save in short-term bonds. In our calibrated model, we also evaluate the extent to which this asymmetric bond position of households matters for term premia quantitatively.

The assumptions in our model are built with solid support from empirical evidence. The first assumption, of a mean-reverting consumption growth process, is prevalent in the macroeconomics literature. The growth of aggregate variables, such as output or consumption, is often decomposed into trend components and cyclical components (business cycles). Such a decomposition is consistent with the mean-reverting assumption. The second assumption, of a segmented market mechanism, is firmly grounded in empirical evidence from the household finance literature. The evidence shows that most households do not purchase most of the assets available to them (Guiso and Sodini, 2012). In fact, the composition of household asset holdings varies greatly across households, even in a developed country such as the United States. Only 50 percent of U.S. households participate in the equity market, according to the 2010 Survey of Consumer Finance (SCF hereafter) data. Moreover, even among the participants in the equity market, many investors still hold low-risk portfolios and do not adjust their portfolios frequently. On the other hand, a small fraction of households actively adjust their portfolios and earn a higher return by taking more aggregate risk. The SCF data also show that a large fraction of households carry mortgage loans and save in short-term safe assets. These households effectively have a long position in short-term bonds and a short position in long-term bonds. As the data also show, wealthier households, a relatively small fraction of all households, tend to hold a higher fraction of long-term bonds in their portfolios.

Only a handful of structural models in the literature are able to deliver an average upward-sloping nominal and/or real yield curve. Many of them modify household preferences into various forms in the standard macroeconomic model. Piazzesi and Schneider (2007) demonstrate that the nominal yield curve can be upward sloping even with a flat or downward-sloping real yield curve since a low-frequency negative correlation between consumption growth and the inflation rate causes inflation risk. They assume a recursive preference, and hence agents are very willing to substitute consumption over time even though they are risk averse. The recursive preference plays a critical role in the low-frequency correlation mattering for the current price. Bansal and Shaliastovich (2013) also generate a positive nominal term premium with inflation risks and recursive preferences. Rudebusch and Swanson (2012) further extend the endowment economy model to a production economy general equilibrium model. By introducing inflation ambiguity into a representative agent model, Ulrich (2013) explains the upward-sloping nominal yield curve with a log utility function. Our work is complementary to the existing papers discussed above since we focus on the real term premia rather than the nominal premia. Wachter (2006) uses a habit-persistence model to explain both the positive real and nominal bond premia. To maintain the consumption level, investors tend to sell long-term bonds during recessions and vice versa during expansions. Namely, the demand for long-term bonds is procyclical, which makes the bond price procyclical and hence the long-term bond itself a risky asset. Rudebusch and Swanson (2012) find that the habit-formation mechanism in Wachter (2006) fails to generate a sizable term premia without distorting the behavior of other macroeconomics variables.

Our benchmark model generates a high and volatile equity premium with a 7.26 percent mean and a 15.63 percent standard deviation, as well as a low and stable risk-free return with a 0.95 mean and a 1.45 percent standard deviation—estimates quite close to those in the asset-pricing literature. Most importantly, our quantitative result also predicts a high real term premium: 1.92 percent for 30-year zero-coupon bonds. This article delivers a reasonable term premium result, with a risk aversion coefficient of 4. For the sensitivity analysis, we further investigate the role of our assumptions by removing each factor one by one.

Our main contribution to the literature is to provide a simple and intuitive story that can reconcile the puzzling disconnect between asset prices, equity and term premia in particular, and aggregate macroeconomic variables. The model in this article integrates the empirical facts of heterogeneous portfolios across households, as found in the household finance literature, and a mean-reverting aggregate consumption process, as found in the macroeconomics literature, to explain the real term-premia puzzle. Our model successfully delivers a positive sign for and significant magnitude of the real term premia. Specifically, we demonstrate the importance of the household portfolio heterogeneity documented in the macro-finance literature, while the majority of asset-pricing models rely on a representative agent framework with modifications to preferences.

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